
We can make a further refinement on the above series by invoking Corollary 2.5 from [1]} in order to write the following
\(G_{\partial M}(x^*,x) &= \frac{1}{2\pi }d_g(x^*,x)^{-1} -\frac{H(x^*)}{4\pi }\log d_h(x^*,x)+\frac{F^3(x^*)}{4\pi }\log d_h(x^*,x)\\ &+\frac{1}{16\pi }\left(\text{II}_{x^*}\left(\frac{\exp _{x^*}^{-1}(x)}{|\exp _{x^*}^{-1}(x)|_h}\right)-\text{II}_{x^*}\left(\frac{\star \exp _{x^*}^{-1}(x)}{|\exp _{x^*}^{-1}(x)|_h}\right)\right)\\&+\frac{1}{4\pi }h_{x^*}\left(F^{\parallel }(x^*),\frac{\exp _{x^*}^{-1}(x)}{|\exp _{x^*}^{-1}(x)|_h}\right) +\Psi ^{-3}(\partial M).\) 
